first integrals
A function $f$ is said to be a first integral of a system (or a constant of motion) if it Poisson-commutes with the Hamiltonian, i.e. ${H, f} = 0$1. This means intuitively that $f(q(t), p(t))$ is constant for all $t$ (if $(q(t), p(t))$ is a solution to the Hamilton's equations).
See also Poisson Bracket.
1
Arnold, V. I. (1989). Mathematical Methods of Classical Mechanics. In Graduate Texts in Mathematics. Springer New York. https://doi.org/10.1007/978-1-4757-2063-1 arnold1989 Mathematical Methods for Classical Mechanics